The necessity was shown in Theorem REF . We now show sufficiency.
Assume that \((2)\)  holds. It suffices to show that \(C_{\phi _1},...,C_{\phi _N}\)  are topologically
d-transitive. Let \(K\)  be a compact subset of \(\Omega \) , \(\varepsilon > 0\) , and \(f\) , \(g_1,...,g_N \in H(\Omega )\) . By
making \(K\)  larger, if needed, we can assume that it is connected, \(\Omega \) -convex and has at
least two holes. According to Lemma REF  and the hypothesis there is some \(n \in \mathbb {N}\)  such that \(\phi _{1_n},...,\phi _{N,n}\)  are injective on a neighborhood of \(K\) ,
\(K\cup \phi _{1,n}(K)\cup \dots \cup \phi _{N,n}(K)\) 
is \(\Omega \) -convex and \(K,\phi _{1,n}(K),...,\phi _{N,n}(K)\)  are pairwise disjoint. As a result, the functions \(g_i\circ \phi _{n}^{-1}\) , as well as \(f\)  on \(K\)  are holomorphic on \(\phi _{i,n}(K)\)  respectively for \(i=1,...,N\) . According to Runge's theorem (see, e.g., [1]}) there exists a function \(h \in H(\Omega )\)  such that
\(|h(z) - f(z)| < \varepsilon \ \textrm {for}\ z \in K,\) 
\(|h(w) - g_i(\phi _{i,n}^{-1}(w))| < \varepsilon \ \textrm {for}\ w \in \phi _{i,n}(K),\) 
hence also
\(|h(\phi _{i,n}(z)) - g(z)| < \varepsilon \ \textrm {for}\ z \in K.\) 
Which implies that \((C_{\phi _{1,n}})_{n=1}^{\infty },...,(C_{\phi _{N,n}})_{n=1}^{\infty }\)  are disjoint transitive.
